Know that a conductor moving across a magnetic field or a changing magnetic field linking with a conductor can induce an e.m.f. in the conductor

Electromagnetic Induction (4.5.1)

Objective (as stated in the Cambridge syllabus)

Know that a conductor moving across a magnetic field or a changing magnetic field linking with a conductor can induce an e.m.f. in the conductor.

Single‑sentence summary

A change in the magnetic flux through a circuit – either because the conductor moves in a static field or because the field itself varies – produces an induced e.m.f., whose direction is given by Lenz’s law.

Key Concepts

• Magnetic flux, \$\Phi\$, is the product of the magnetic field strength, the area it penetrates and the cosine of the angle between the field and the area normal:

  \$\Phi = B A \cos\theta\$

• Any change in \$\Phi\$ (by altering \$B\$, \$A\$, or \$\theta\$) induces an e.m.f. according to Faraday’s law.
• Two practical ways of changing flux:

  1. Moving a conductor (or part of a circuit) through a magnetic field.
  2. Changing the magnetic field that links a stationary conductor.

• The induced e.m.f. always acts so as to oppose the change that produced it (Lenz’s law).

Faraday’s Law of Electromagnetic Induction

Negative sign = Lenz’s law (direction of the e.m.f.).

Induced e.m.f. in a Moving Conductor

For a straight conductor of length \$L\$ moving with velocity \$v\$ in a uniform magnetic field \$B\$:

\$\mathcal{E}= B L v \sin\theta\$

• \$\theta\$ = angle between the direction of motion and the magnetic field.
• Maximum e.m.f. when \$\theta = 90^{\circ}\$ (motion ⟂ field).
• Direction of the induced current is found with the right‑hand rule for generators (see Lenz’s‑law section).

Induced e.m.f. in a Changing Magnetic Field

For a coil of \$N\$ turns whose flux changes:

\$\mathcal{E}= -N\frac{d\Phi}{dt}= -N\frac{d}{dt}(B A \cos\theta)\$

Any change in \$B\$, \$A\$ or \$\theta\$ produces an e.m.f.

Factors that Influence the Magnitude of the Induced e.m.f.

Lenz’s Law (Direction of the Induced e.m.f.)

The induced current creates a magnetic field that opposes the original change in flux. The direction can be obtained with the right‑hand generator rule:

1. Point the thumb in the direction of the conductor’s motion (or the direction of increasing area).
2. Point the first finger in the direction of the magnetic field \$\mathbf{B}\$.
3. The middle finger, perpendicular to the other two, gives the direction of conventional current (and thus the polarity of the induced e.m.f.).

Experiment to Demonstrate Electromagnetic Induction

Apparatus: strong bar magnet, straight copper rod (or rectangular coil), wooden rails, pulley, galvanometer (or voltmeter), ruler, support stand.

Procedure (moving‑conductor method):

1. Fix the rails so the copper rod can slide without friction.
2. Attach the magnet to a pulley so it can fall vertically past the rod; the magnet’s north‑south axis should be vertical.
3. Connect the ends of the rod to the galvanometer.
4. Release the magnet and observe a brief deflection as it passes the rod.
5. Repeat with the magnet moving upward; the deflection reverses, confirming Lenz’s law.
6. Vary the drop height to change the speed – larger speed gives a larger deflection (illustrating the \$v\$ dependence).

Observation: The magnitude of the galvanometer deflection is proportional to the speed of the magnet and to the strength of the magnetic field.

Suggested Diagrams (placeholders)

Worked Example Problems

1. Problem: A rod 0.15 m long moves at 4.0 m s⁻¹ through a magnetic field of 0.25 T directed into the page. Find the magnitude of the induced e.m.f.

   Solution:

   \$\mathcal{E}= B L v = (0.25\;\text{T})(0.15\;\text{m})(4.0\;\text{m s}^{-1}) = 0.15\;\text{V}\$

2. Problem: A single loop of area \$0.02\;\text{m}^2\$ is in a uniform magnetic field that increases from \$0.30\;\text{T}\$ to \$0.50\;\text{T}\$ in \$0.10\;\text{s}\$. Calculate the average induced e.m.f.

   Solution:

   \$\Delta\Phi = A\Delta B = (0.02)(0.50-0.30)=0.004\;\text{Wb}\$

   \$\mathcal{E}_{\text{avg}} = -\frac{\Delta\Phi}{\Delta t}= -\frac{0.004}{0.10}= -0.040\;\text{V}\$

   Magnitude = \$0.040\;\text{V}\$.

3. Problem: A coil of 200 turns has a radius of \$5\;\text{cm}\$ and is placed in a magnetic field that is decreasing at a rate of \$2.0\times10^{-3}\;\text{T s}^{-1}\$. Find the induced e.m.f.

   Solution:

   \$A = \pi r^{2}= \pi(0.05)^{2}=7.85\times10^{-3}\;\text{m}^{2}\$

   \$\mathcal{E}= -N A \frac{dB}{dt}= -(200)(7.85\times10^{-3})(-2.0\times10^{-3})=3.14\times10^{-3}\;\text{V}\$

Common Misconceptions

• “Only a changing magnetic field can induce an e.m.f.” – Motion of a conductor through a static field also changes the linked flux.
• “The induced e.m.f. is always in the same direction as the motion.” – Its direction is set by Lenz’s law, not by motion alone.
• “A larger coil always gives a larger e.m.f.” – The coil must experience a change in flux; a larger area with no change produces no e.m.f.
• “The minus sign in Faraday’s law is just a convention.” – It embodies Lenz’s law and is essential for predicting the correct polarity.

Summary Checklist (Cambridge 0625 – 4.5.1)

• State the syllabus objective verbatim.
• Explain that an induced e.m.f. arises when magnetic flux through a conductor changes (by motion or by a varying field).
• List the factors that affect the magnitude of \$\mathcal{E}\$ and recognise they are all directly proportional.
• Describe a simple experiment (magnet‑rod) that demonstrates electromagnetic induction.
• State Lenz’s law exactly: “The induced e.m.f. always opposes the change that produced it.”
• Use correct terminology: e.m.f., magnetic flux, induced current, polarity, etc.

Summary

• Induced e.m.f. appears whenever the magnetic flux linking a conductor changes.
• Flux can be altered by moving the conductor in a magnetic field or by varying the magnetic field itself.
• Faraday’s law quantifies the e.m.f.: \$\displaystyle \mathcal{E}= -\frac{d\Phi}{dt}\$ (or \$-N\frac{d\Phi}{dt}\$ for \$N\$ turns).
• Lenz’s law gives the direction – the induced current creates a magnetic field that opposes the original change.
• The magnitude depends on \$B\$, \$L\$, \$v\$, \$N\$, \$A\$, the relevant angle and the rate of change of \$B\$ (or \$A\$, \$\theta\$).

Practice Questions

1. A rectangular loop (width \$0.10\;\text{m}\$, height \$0.20\;\text{m}\$) is pulled out of a uniform magnetic field of \$0.40\;\text{T}\$ at a constant speed of \$2.0\;\text{m s}^{-1}\$. Calculate the induced e.m.f. while the loop is leaving the field.

2. A solenoid with \$500\$ turns and cross‑sectional area \$1.5\times10^{-3}\;\text{m}^{2}\$ has its current switched off uniformly over \$0.25\;\text{s}\$. If the initial magnetic field inside the solenoid is \$0.80\;\text{T}\$, find the average induced e.m.f. in the coil.

3. Explain, using Lenz’s law, why a metal disc rotating in a magnetic field can be used as a generator.